\(\int \frac {3+3 \sin (e+f x)}{(c+d \sin (e+f x))^{3/2}} \, dx\) [486]

Optimal result

Integrand size = 25, antiderivative size = 166 \[ \int \frac {3+3 \sin (e+f x)}{(c+d \sin (e+f x))^{3/2}} \, dx=-\frac {6 \cos (e+f x)}{(c+d) f \sqrt {c+d \sin (e+f x)}}-\frac {6 E\left (\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right )|\frac {2 d}{c+d}\right ) \sqrt {c+d \sin (e+f x)}}{d (c+d) f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}+\frac {6 \operatorname {EllipticF}\left (\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right ),\frac {2 d}{c+d}\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}{d f \sqrt {c+d \sin (e+f x)}} \]

[Out]

-2*a*cos(f*x+e)/(c+d)/f/(c+d*sin(f*x+e))^(1/2)+2*a*(sin(1/2*e+1/4*Pi+1/2*f*x)^2)^(1/2)/sin(1/2*e+1/4*Pi+1/2*f*
x)*EllipticE(cos(1/2*e+1/4*Pi+1/2*f*x),2^(1/2)*(d/(c+d))^(1/2))*(c+d*sin(f*x+e))^(1/2)/d/(c+d)/f/((c+d*sin(f*x
+e))/(c+d))^(1/2)-2*a*(sin(1/2*e+1/4*Pi+1/2*f*x)^2)^(1/2)/sin(1/2*e+1/4*Pi+1/2*f*x)*EllipticF(cos(1/2*e+1/4*Pi
+1/2*f*x),2^(1/2)*(d/(c+d))^(1/2))*((c+d*sin(f*x+e))/(c+d))^(1/2)/d/f/(c+d*sin(f*x+e))^(1/2)

Rubi [A] (verified)

Time = 0.16 (sec) , antiderivative size = 169, normalized size of antiderivative = 1.02, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.240, Rules used = {2833, 2831, 2742, 2740, 2734, 2732} \[ \int \frac {3+3 \sin (e+f x)}{(c+d \sin (e+f x))^{3/2}} \, dx=-\frac {2 a \cos (e+f x)}{f (c+d) \sqrt {c+d \sin (e+f x)}}+\frac {2 a \sqrt {\frac {c+d \sin (e+f x)}{c+d}} \operatorname {EllipticF}\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right ),\frac {2 d}{c+d}\right )}{d f \sqrt {c+d \sin (e+f x)}}-\frac {2 a \sqrt {c+d \sin (e+f x)} E\left (\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )|\frac {2 d}{c+d}\right )}{d f (c+d) \sqrt {\frac {c+d \sin (e+f x)}{c+d}}} \]

[In]

Int[(a + a*Sin[e + f*x])/(c + d*Sin[e + f*x])^(3/2),x]

[Out]

(-2*a*Cos[e + f*x])/((c + d)*f*Sqrt[c + d*Sin[e + f*x]]) - (2*a*EllipticE[(e - Pi/2 + f*x)/2, (2*d)/(c + d)]*S
qrt[c + d*Sin[e + f*x]])/(d*(c + d)*f*Sqrt[(c + d*Sin[e + f*x])/(c + d)]) + (2*a*EllipticF[(e - Pi/2 + f*x)/2,
(2*d)/(c + d)]*Sqrt[(c + d*Sin[e + f*x])/(c + d)])/(d*f*Sqrt[c + d*Sin[e + f*x]])

Rule 2732

Int[Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[2*(Sqrt[a + b]/d)*EllipticE[(1/2)*(c - Pi/2
+ d*x), 2*(b/(a + b))], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2, 0] && GtQ[a + b, 0]

Rule 2734

Int[Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Dist[Sqrt[a + b*Sin[c + d*x]]/Sqrt[(a + b*Sin[c +
d*x])/(a + b)], Int[Sqrt[a/(a + b) + (b/(a + b))*Sin[c + d*x]], x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 -
b^2, 0] &&  !GtQ[a + b, 0]

Rule 2740

Int[1/Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2/(d*Sqrt[a + b]))*EllipticF[(1/2)*(c - P
i/2 + d*x), 2*(b/(a + b))], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2, 0] && GtQ[a + b, 0]

Rule 2742

Int[1/Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Dist[Sqrt[(a + b*Sin[c + d*x])/(a + b)]/Sqrt[a
+ b*Sin[c + d*x]], Int[1/Sqrt[a/(a + b) + (b/(a + b))*Sin[c + d*x]], x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a
^2 - b^2, 0] &&  !GtQ[a + b, 0]

Rule 2831

Int[((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])/Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]], x_Symbol] :> Dist[(b*c
- a*d)/b, Int[1/Sqrt[a + b*Sin[e + f*x]], x], x] + Dist[d/b, Int[Sqrt[a + b*Sin[e + f*x]], x], x] /; FreeQ[{a
, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0]

Rule 2833

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(-(
b*c - a*d))*Cos[e + f*x]*((a + b*Sin[e + f*x])^(m + 1)/(f*(m + 1)*(a^2 - b^2))), x] + Dist[1/((m + 1)*(a^2 - b
^2)), Int[(a + b*Sin[e + f*x])^(m + 1)*Simp[(a*c - b*d)*(m + 1) - (b*c - a*d)*(m + 2)*Sin[e + f*x], x], x], x]
/; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && LtQ[m, -1] && IntegerQ[2*m]

Rubi steps \begin{align*} \text {integral}& = -\frac {2 a \cos (e+f x)}{(c+d) f \sqrt {c+d \sin (e+f x)}}-\frac {2 \int \frac {-\frac {1}{2} a (c-d)+\frac {1}{2} a (c-d) \sin (e+f x)}{\sqrt {c+d \sin (e+f x)}} \, dx}{c^2-d^2} \\ & = -\frac {2 a \cos (e+f x)}{(c+d) f \sqrt {c+d \sin (e+f x)}}+\frac {a \int \frac {1}{\sqrt {c+d \sin (e+f x)}} \, dx}{d}-\frac {a \int \sqrt {c+d \sin (e+f x)} \, dx}{d (c+d)} \\ & = -\frac {2 a \cos (e+f x)}{(c+d) f \sqrt {c+d \sin (e+f x)}}-\frac {\left (a \sqrt {c+d \sin (e+f x)}\right ) \int \sqrt {\frac {c}{c+d}+\frac {d \sin (e+f x)}{c+d}} \, dx}{d (c+d) \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}+\frac {\left (a \sqrt {\frac {c+d \sin (e+f x)}{c+d}}\right ) \int \frac {1}{\sqrt {\frac {c}{c+d}+\frac {d \sin (e+f x)}{c+d}}} \, dx}{d \sqrt {c+d \sin (e+f x)}} \\ & = -\frac {2 a \cos (e+f x)}{(c+d) f \sqrt {c+d \sin (e+f x)}}-\frac {2 a E\left (\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right )|\frac {2 d}{c+d}\right ) \sqrt {c+d \sin (e+f x)}}{d (c+d) f \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}+\frac {2 a \operatorname {EllipticF}\left (\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right ),\frac {2 d}{c+d}\right ) \sqrt {\frac {c+d \sin (e+f x)}{c+d}}}{d f \sqrt {c+d \sin (e+f x)}} \\ \end{align*}

Mathematica [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 6 vs. order 4 in optimal.

Time = 6.66 (sec) , antiderivative size = 938, normalized size of antiderivative = 5.65 \[ \int \frac {3+3 \sin (e+f x)}{(c+d \sin (e+f x))^{3/2}} \, dx=3 \left (\frac {(1+\sin (e+f x)) \sqrt {c+d \sin (e+f x)} \left (-\frac {2 \csc (e) \sec (e)}{d (c+d) f}+\frac {2 \csc (e) (c \cos (e)+d \sin (f x))}{d (c+d) f (c+d \sin (e+f x))}\right )}{\left (\cos \left (\frac {e}{2}+\frac {f x}{2}\right )+\sin \left (\frac {e}{2}+\frac {f x}{2}\right )\right )^2}-\frac {\sec (e) (1+\sin (e+f x)) \left (-\frac {\operatorname {AppellF1}\left (-\frac {1}{2},-\frac {1}{2},-\frac {1}{2},\frac {1}{2},-\frac {\csc (e) \left (c+d \cos (f x-\arctan (\cot (e))) \sqrt {1+\cot ^2(e)} \sin (e)\right )}{d \sqrt {1+\cot ^2(e)} \left (1-\frac {c \csc (e)}{d \sqrt {1+\cot ^2(e)}}\right )},-\frac {\csc (e) \left (c+d \cos (f x-\arctan (\cot (e))) \sqrt {1+\cot ^2(e)} \sin (e)\right )}{d \sqrt {1+\cot ^2(e)} \left (-1-\frac {c \csc (e)}{d \sqrt {1+\cot ^2(e)}}\right )}\right ) \cot (e) \sin (f x-\arctan (\cot (e)))}{\sqrt {1+\cot ^2(e)} \sqrt {\frac {d \sqrt {1+\cot ^2(e)}+d \cos (f x-\arctan (\cot (e))) \sqrt {1+\cot ^2(e)}}{d \sqrt {1+\cot ^2(e)}-c \csc (e)}} \sqrt {\frac {d \sqrt {1+\cot ^2(e)}-d \cos (f x-\arctan (\cot (e))) \sqrt {1+\cot ^2(e)}}{d \sqrt {1+\cot ^2(e)}+c \csc (e)}} \sqrt {c+d \cos (f x-\arctan (\cot (e))) \sqrt {1+\cot ^2(e)} \sin (e)}}-\frac {\frac {2 d \sin (e) \left (c+d \cos (f x-\arctan (\cot (e))) \sqrt {1+\cot ^2(e)} \sin (e)\right )}{d^2 \cos ^2(e)+d^2 \sin ^2(e)}-\frac {\cot (e) \sin (f x-\arctan (\cot (e)))}{\sqrt {1+\cot ^2(e)}}}{\sqrt {c+d \cos (f x-\arctan (\cot (e))) \sqrt {1+\cot ^2(e)} \sin (e)}}\right )}{(c+d) f \left (\cos \left (\frac {e}{2}+\frac {f x}{2}\right )+\sin \left (\frac {e}{2}+\frac {f x}{2}\right )\right )^2}+\frac {2 \operatorname {AppellF1}\left (\frac {1}{2},\frac {1}{2},\frac {1}{2},\frac {3}{2},-\frac {\sec (e) \left (c+d \cos (e) \sin (f x+\arctan (\tan (e))) \sqrt {1+\tan ^2(e)}\right )}{d \sqrt {1+\tan ^2(e)} \left (1-\frac {c \sec (e)}{d \sqrt {1+\tan ^2(e)}}\right )},-\frac {\sec (e) \left (c+d \cos (e) \sin (f x+\arctan (\tan (e))) \sqrt {1+\tan ^2(e)}\right )}{d \sqrt {1+\tan ^2(e)} \left (-1-\frac {c \sec (e)}{d \sqrt {1+\tan ^2(e)}}\right )}\right ) \sec (e) \sec (f x+\arctan (\tan (e))) (1+\sin (e+f x)) \sqrt {\frac {d \sqrt {1+\tan ^2(e)}-d \sin (f x+\arctan (\tan (e))) \sqrt {1+\tan ^2(e)}}{c \sec (e)+d \sqrt {1+\tan ^2(e)}}} \sqrt {\frac {d \sqrt {1+\tan ^2(e)}+d \sin (f x+\arctan (\tan (e))) \sqrt {1+\tan ^2(e)}}{-c \sec (e)+d \sqrt {1+\tan ^2(e)}}} \sqrt {c+d \cos (e) \sin (f x+\arctan (\tan (e))) \sqrt {1+\tan ^2(e)}}}{d (c+d) f \left (\cos \left (\frac {e}{2}+\frac {f x}{2}\right )+\sin \left (\frac {e}{2}+\frac {f x}{2}\right )\right )^2 \sqrt {1+\tan ^2(e)}}\right ) \]

[In]

Integrate[(3 + 3*Sin[e + f*x])/(c + d*Sin[e + f*x])^(3/2),x]

[Out]

3*(((1 + Sin[e + f*x])*Sqrt[c + d*Sin[e + f*x]]*((-2*Csc[e]*Sec[e])/(d*(c + d)*f) + (2*Csc[e]*(c*Cos[e] + d*Si
n[f*x]))/(d*(c + d)*f*(c + d*Sin[e + f*x]))))/(Cos[e/2 + (f*x)/2] + Sin[e/2 + (f*x)/2])^2 - (Sec[e]*(1 + Sin[e
+ f*x])*(-((AppellF1[-1/2, -1/2, -1/2, 1/2, -((Csc[e]*(c + d*Cos[f*x - ArcTan[Cot[e]]]*Sqrt[1 + Cot[e]^2]*Sin
[e]))/(d*Sqrt[1 + Cot[e]^2]*(1 - (c*Csc[e])/(d*Sqrt[1 + Cot[e]^2])))), -((Csc[e]*(c + d*Cos[f*x - ArcTan[Cot[e
]]]*Sqrt[1 + Cot[e]^2]*Sin[e]))/(d*Sqrt[1 + Cot[e]^2]*(-1 - (c*Csc[e])/(d*Sqrt[1 + Cot[e]^2]))))]*Cot[e]*Sin[f
*x - ArcTan[Cot[e]]])/(Sqrt[1 + Cot[e]^2]*Sqrt[(d*Sqrt[1 + Cot[e]^2] + d*Cos[f*x - ArcTan[Cot[e]]]*Sqrt[1 + Co
t[e]^2])/(d*Sqrt[1 + Cot[e]^2] - c*Csc[e])]*Sqrt[(d*Sqrt[1 + Cot[e]^2] - d*Cos[f*x - ArcTan[Cot[e]]]*Sqrt[1 +
Cot[e]^2])/(d*Sqrt[1 + Cot[e]^2] + c*Csc[e])]*Sqrt[c + d*Cos[f*x - ArcTan[Cot[e]]]*Sqrt[1 + Cot[e]^2]*Sin[e]])
) - ((2*d*Sin[e]*(c + d*Cos[f*x - ArcTan[Cot[e]]]*Sqrt[1 + Cot[e]^2]*Sin[e]))/(d^2*Cos[e]^2 + d^2*Sin[e]^2) -
(Cot[e]*Sin[f*x - ArcTan[Cot[e]]])/Sqrt[1 + Cot[e]^2])/Sqrt[c + d*Cos[f*x - ArcTan[Cot[e]]]*Sqrt[1 + Cot[e]^2]
*Sin[e]]))/((c + d)*f*(Cos[e/2 + (f*x)/2] + Sin[e/2 + (f*x)/2])^2) + (2*AppellF1[1/2, 1/2, 1/2, 3/2, -((Sec[e]
*(c + d*Cos[e]*Sin[f*x + ArcTan[Tan[e]]]*Sqrt[1 + Tan[e]^2]))/(d*Sqrt[1 + Tan[e]^2]*(1 - (c*Sec[e])/(d*Sqrt[1
+ Tan[e]^2])))), -((Sec[e]*(c + d*Cos[e]*Sin[f*x + ArcTan[Tan[e]]]*Sqrt[1 + Tan[e]^2]))/(d*Sqrt[1 + Tan[e]^2]*
(-1 - (c*Sec[e])/(d*Sqrt[1 + Tan[e]^2]))))]*Sec[e]*Sec[f*x + ArcTan[Tan[e]]]*(1 + Sin[e + f*x])*Sqrt[(d*Sqrt[1
+ Tan[e]^2] - d*Sin[f*x + ArcTan[Tan[e]]]*Sqrt[1 + Tan[e]^2])/(c*Sec[e] + d*Sqrt[1 + Tan[e]^2])]*Sqrt[(d*Sqrt
[1 + Tan[e]^2] + d*Sin[f*x + ArcTan[Tan[e]]]*Sqrt[1 + Tan[e]^2])/(-(c*Sec[e]) + d*Sqrt[1 + Tan[e]^2])]*Sqrt[c
+ d*Cos[e]*Sin[f*x + ArcTan[Tan[e]]]*Sqrt[1 + Tan[e]^2]])/(d*(c + d)*f*(Cos[e/2 + (f*x)/2] + Sin[e/2 + (f*x)/2
])^2*Sqrt[1 + Tan[e]^2]))

Maple [A] (verified)

Time = 2.57 (sec) , antiderivative size = 246, normalized size of antiderivative = 1.48

method	result	size
default	\(\frac {2 \left (\sqrt {\frac {c +d \sin \left (f x +e \right )}{c -d}}\, \sqrt {-\frac {\left (\sin \left (f x +e \right )-1\right ) d}{c +d}}\, \sqrt {-\frac {d \left (\sin \left (f x +e \right )+1\right )}{c -d}}\, E\left (\sqrt {\frac {c +d \sin \left (f x +e \right )}{c -d}}, \sqrt {\frac {c -d}{c +d}}\right ) c^{2}-\sqrt {\frac {c +d \sin \left (f x +e \right )}{c -d}}\, \sqrt {-\frac {\left (\sin \left (f x +e \right )-1\right ) d}{c +d}}\, \sqrt {-\frac {d \left (\sin \left (f x +e \right )+1\right )}{c -d}}\, E\left (\sqrt {\frac {c +d \sin \left (f x +e \right )}{c -d}}, \sqrt {\frac {c -d}{c +d}}\right ) d^{2}+\left (\sin ^{2}\left (f x +e \right )\right ) d^{2}-d^{2}\right ) a}{d^{2} \left (c +d \right ) \cos \left (f x +e \right ) \sqrt {c +d \sin \left (f x +e \right )}\, f}\)	\(246\)
parts	\(\frac {2 a \left (c^{2} \sqrt {\frac {c +d \sin \left (f x +e \right )}{c -d}}\, \sqrt {-\frac {\left (\sin \left (f x +e \right )-1\right ) d}{c +d}}\, \sqrt {-\frac {d \left (\sin \left (f x +e \right )+1\right )}{c -d}}\, F\left (\sqrt {\frac {c +d \sin \left (f x +e \right )}{c -d}}, \sqrt {\frac {c -d}{c +d}}\right )-\sqrt {\frac {c +d \sin \left (f x +e \right )}{c -d}}\, \sqrt {-\frac {\left (\sin \left (f x +e \right )-1\right ) d}{c +d}}\, \sqrt {-\frac {d \left (\sin \left (f x +e \right )+1\right )}{c -d}}\, F\left (\sqrt {\frac {c +d \sin \left (f x +e \right )}{c -d}}, \sqrt {\frac {c -d}{c +d}}\right ) d^{2}-\sqrt {\frac {c +d \sin \left (f x +e \right )}{c -d}}\, \sqrt {-\frac {\left (\sin \left (f x +e \right )-1\right ) d}{c +d}}\, \sqrt {-\frac {d \left (\sin \left (f x +e \right )+1\right )}{c -d}}\, E\left (\sqrt {\frac {c +d \sin \left (f x +e \right )}{c -d}}, \sqrt {\frac {c -d}{c +d}}\right ) c^{2}+\sqrt {\frac {c +d \sin \left (f x +e \right )}{c -d}}\, \sqrt {-\frac {\left (\sin \left (f x +e \right )-1\right ) d}{c +d}}\, \sqrt {-\frac {d \left (\sin \left (f x +e \right )+1\right )}{c -d}}\, E\left (\sqrt {\frac {c +d \sin \left (f x +e \right )}{c -d}}, \sqrt {\frac {c -d}{c +d}}\right ) d^{2}-\left (\sin ^{2}\left (f x +e \right )\right ) d^{2}+d^{2}\right )}{d \left (c^{2}-d^{2}\right ) \cos \left (f x +e \right ) \sqrt {c +d \sin \left (f x +e \right )}\, f}-\frac {2 a \left (\sqrt {\frac {c +d \sin \left (f x +e \right )}{c -d}}\, \sqrt {-\frac {\left (\sin \left (f x +e \right )-1\right ) d}{c +d}}\, \sqrt {-\frac {d \left (\sin \left (f x +e \right )+1\right )}{c -d}}\, F\left (\sqrt {\frac {c +d \sin \left (f x +e \right )}{c -d}}, \sqrt {\frac {c -d}{c +d}}\right ) c^{2} d -\sqrt {\frac {c +d \sin \left (f x +e \right )}{c -d}}\, \sqrt {-\frac {\left (\sin \left (f x +e \right )-1\right ) d}{c +d}}\, \sqrt {-\frac {d \left (\sin \left (f x +e \right )+1\right )}{c -d}}\, F\left (\sqrt {\frac {c +d \sin \left (f x +e \right )}{c -d}}, \sqrt {\frac {c -d}{c +d}}\right ) d^{3}-\sqrt {\frac {c +d \sin \left (f x +e \right )}{c -d}}\, \sqrt {-\frac {\left (\sin \left (f x +e \right )-1\right ) d}{c +d}}\, \sqrt {-\frac {d \left (\sin \left (f x +e \right )+1\right )}{c -d}}\, E\left (\sqrt {\frac {c +d \sin \left (f x +e \right )}{c -d}}, \sqrt {\frac {c -d}{c +d}}\right ) c^{3}+\sqrt {\frac {c +d \sin \left (f x +e \right )}{c -d}}\, \sqrt {-\frac {\left (\sin \left (f x +e \right )-1\right ) d}{c +d}}\, \sqrt {-\frac {d \left (\sin \left (f x +e \right )+1\right )}{c -d}}\, E\left (\sqrt {\frac {c +d \sin \left (f x +e \right )}{c -d}}, \sqrt {\frac {c -d}{c +d}}\right ) c \,d^{2}-\left (\sin ^{2}\left (f x +e \right )\right ) c \,d^{2}+c \,d^{2}\right )}{d^{2} \left (c^{2}-d^{2}\right ) \cos \left (f x +e \right ) \sqrt {c +d \sin \left (f x +e \right )}\, f}\)	\(893\)

[In]

int((a+a*sin(f*x+e))/(c+d*sin(f*x+e))^(3/2),x,method=_RETURNVERBOSE)

[Out]

2*(((c+d*sin(f*x+e))/(c-d))^(1/2)*(-(sin(f*x+e)-1)*d/(c+d))^(1/2)*(-d*(sin(f*x+e)+1)/(c-d))^(1/2)*EllipticE(((
c+d*sin(f*x+e))/(c-d))^(1/2),((c-d)/(c+d))^(1/2))*c^2-((c+d*sin(f*x+e))/(c-d))^(1/2)*(-(sin(f*x+e)-1)*d/(c+d))
^(1/2)*(-d*(sin(f*x+e)+1)/(c-d))^(1/2)*EllipticE(((c+d*sin(f*x+e))/(c-d))^(1/2),((c-d)/(c+d))^(1/2))*d^2+sin(f
*x+e)^2*d^2-d^2)/d^2*a/(c+d)/cos(f*x+e)/(c+d*sin(f*x+e))^(1/2)/f

Fricas [C] (verification not implemented)

Result contains higher order function than in optimal. Order 9 vs. order 4.

Time = 0.11 (sec) , antiderivative size = 528, normalized size of antiderivative = 3.18 \[ \int \frac {3+3 \sin (e+f x)}{(c+d \sin (e+f x))^{3/2}} \, dx=-\frac {6 \, \sqrt {d \sin \left (f x + e\right ) + c} a d^{2} \cos \left (f x + e\right ) - {\left (\sqrt {2} {\left (2 \, a c d + 3 \, a d^{2}\right )} \sin \left (f x + e\right ) + \sqrt {2} {\left (2 \, a c^{2} + 3 \, a c d\right )}\right )} \sqrt {i \, d} {\rm weierstrassPInverse}\left (-\frac {4 \, {\left (4 \, c^{2} - 3 \, d^{2}\right )}}{3 \, d^{2}}, -\frac {8 \, {\left (8 i \, c^{3} - 9 i \, c d^{2}\right )}}{27 \, d^{3}}, \frac {3 \, d \cos \left (f x + e\right ) - 3 i \, d \sin \left (f x + e\right ) - 2 i \, c}{3 \, d}\right ) - {\left (\sqrt {2} {\left (2 \, a c d + 3 \, a d^{2}\right )} \sin \left (f x + e\right ) + \sqrt {2} {\left (2 \, a c^{2} + 3 \, a c d\right )}\right )} \sqrt {-i \, d} {\rm weierstrassPInverse}\left (-\frac {4 \, {\left (4 \, c^{2} - 3 \, d^{2}\right )}}{3 \, d^{2}}, -\frac {8 \, {\left (-8 i \, c^{3} + 9 i \, c d^{2}\right )}}{27 \, d^{3}}, \frac {3 \, d \cos \left (f x + e\right ) + 3 i \, d \sin \left (f x + e\right ) + 2 i \, c}{3 \, d}\right ) + 3 \, {\left (-i \, \sqrt {2} a d^{2} \sin \left (f x + e\right ) - i \, \sqrt {2} a c d\right )} \sqrt {i \, d} {\rm weierstrassZeta}\left (-\frac {4 \, {\left (4 \, c^{2} - 3 \, d^{2}\right )}}{3 \, d^{2}}, -\frac {8 \, {\left (8 i \, c^{3} - 9 i \, c d^{2}\right )}}{27 \, d^{3}}, {\rm weierstrassPInverse}\left (-\frac {4 \, {\left (4 \, c^{2} - 3 \, d^{2}\right )}}{3 \, d^{2}}, -\frac {8 \, {\left (8 i \, c^{3} - 9 i \, c d^{2}\right )}}{27 \, d^{3}}, \frac {3 \, d \cos \left (f x + e\right ) - 3 i \, d \sin \left (f x + e\right ) - 2 i \, c}{3 \, d}\right )\right ) + 3 \, {\left (i \, \sqrt {2} a d^{2} \sin \left (f x + e\right ) + i \, \sqrt {2} a c d\right )} \sqrt {-i \, d} {\rm weierstrassZeta}\left (-\frac {4 \, {\left (4 \, c^{2} - 3 \, d^{2}\right )}}{3 \, d^{2}}, -\frac {8 \, {\left (-8 i \, c^{3} + 9 i \, c d^{2}\right )}}{27 \, d^{3}}, {\rm weierstrassPInverse}\left (-\frac {4 \, {\left (4 \, c^{2} - 3 \, d^{2}\right )}}{3 \, d^{2}}, -\frac {8 \, {\left (-8 i \, c^{3} + 9 i \, c d^{2}\right )}}{27 \, d^{3}}, \frac {3 \, d \cos \left (f x + e\right ) + 3 i \, d \sin \left (f x + e\right ) + 2 i \, c}{3 \, d}\right )\right )}{3 \, {\left ({\left (c d^{3} + d^{4}\right )} f \sin \left (f x + e\right ) + {\left (c^{2} d^{2} + c d^{3}\right )} f\right )}} \]

[In]

integrate((a+a*sin(f*x+e))/(c+d*sin(f*x+e))^(3/2),x, algorithm="fricas")

[Out]

-1/3*(6*sqrt(d*sin(f*x + e) + c)*a*d^2*cos(f*x + e) - (sqrt(2)*(2*a*c*d + 3*a*d^2)*sin(f*x + e) + sqrt(2)*(2*a
*c^2 + 3*a*c*d))*sqrt(I*d)*weierstrassPInverse(-4/3*(4*c^2 - 3*d^2)/d^2, -8/27*(8*I*c^3 - 9*I*c*d^2)/d^3, 1/3*
(3*d*cos(f*x + e) - 3*I*d*sin(f*x + e) - 2*I*c)/d) - (sqrt(2)*(2*a*c*d + 3*a*d^2)*sin(f*x + e) + sqrt(2)*(2*a*
c^2 + 3*a*c*d))*sqrt(-I*d)*weierstrassPInverse(-4/3*(4*c^2 - 3*d^2)/d^2, -8/27*(-8*I*c^3 + 9*I*c*d^2)/d^3, 1/3
*(3*d*cos(f*x + e) + 3*I*d*sin(f*x + e) + 2*I*c)/d) + 3*(-I*sqrt(2)*a*d^2*sin(f*x + e) - I*sqrt(2)*a*c*d)*sqrt
(I*d)*weierstrassZeta(-4/3*(4*c^2 - 3*d^2)/d^2, -8/27*(8*I*c^3 - 9*I*c*d^2)/d^3, weierstrassPInverse(-4/3*(4*c
^2 - 3*d^2)/d^2, -8/27*(8*I*c^3 - 9*I*c*d^2)/d^3, 1/3*(3*d*cos(f*x + e) - 3*I*d*sin(f*x + e) - 2*I*c)/d)) + 3*
(I*sqrt(2)*a*d^2*sin(f*x + e) + I*sqrt(2)*a*c*d)*sqrt(-I*d)*weierstrassZeta(-4/3*(4*c^2 - 3*d^2)/d^2, -8/27*(-
8*I*c^3 + 9*I*c*d^2)/d^3, weierstrassPInverse(-4/3*(4*c^2 - 3*d^2)/d^2, -8/27*(-8*I*c^3 + 9*I*c*d^2)/d^3, 1/3*
(3*d*cos(f*x + e) + 3*I*d*sin(f*x + e) + 2*I*c)/d)))/((c*d^3 + d^4)*f*sin(f*x + e) + (c^2*d^2 + c*d^3)*f)

Sympy [F(-1)]

Timed out. \[ \int \frac {3+3 \sin (e+f x)}{(c+d \sin (e+f x))^{3/2}} \, dx=\text {Timed out} \]

[In]

integrate((a+a*sin(f*x+e))/(c+d*sin(f*x+e))**(3/2),x)

[Out]

Timed out

Maxima [F]

\[ \int \frac {3+3 \sin (e+f x)}{(c+d \sin (e+f x))^{3/2}} \, dx=\int { \frac {a \sin \left (f x + e\right ) + a}{{\left (d \sin \left (f x + e\right ) + c\right )}^{\frac {3}{2}}} \,d x } \]

[In]

integrate((a+a*sin(f*x+e))/(c+d*sin(f*x+e))^(3/2),x, algorithm="maxima")

[Out]

integrate((a*sin(f*x + e) + a)/(d*sin(f*x + e) + c)^(3/2), x)

Giac [F]

\[ \int \frac {3+3 \sin (e+f x)}{(c+d \sin (e+f x))^{3/2}} \, dx=\int { \frac {a \sin \left (f x + e\right ) + a}{{\left (d \sin \left (f x + e\right ) + c\right )}^{\frac {3}{2}}} \,d x } \]

[In]

integrate((a+a*sin(f*x+e))/(c+d*sin(f*x+e))^(3/2),x, algorithm="giac")

[Out]

integrate((a*sin(f*x + e) + a)/(d*sin(f*x + e) + c)^(3/2), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {3+3 \sin (e+f x)}{(c+d \sin (e+f x))^{3/2}} \, dx=\int \frac {a+a\,\sin \left (e+f\,x\right )}{{\left (c+d\,\sin \left (e+f\,x\right )\right )}^{3/2}} \,d x \]

[In]

int((a + a*sin(e + f*x))/(c + d*sin(e + f*x))^(3/2),x)

[Out]

int((a + a*sin(e + f*x))/(c + d*sin(e + f*x))^(3/2), x)